左 means ‘left’. Its part 工 is ‘work’ and the remaining component looks like ‘head, shoulders plus arm’. Do you often use the ‘left’ arm or hand to work? Maybe, our ancestors believed that the left hand was for the main labor job in daily life.
Then the 右 （right）hand is used to serve the mouth at the dining table.
In general,the structure of 左右 tells us that our hands work hard for food! (Human beings could make a living with our own hands!------In other words, “ Laziness would starve you to death!’)

In movie, we often see the boss sitting in middle with his men standing on both of the left-and- right sides. The former could fully control the latter. So sometime the phrase ‘左右’ means ‘controling’ or ‘ruling’.

The ancient image of “左” is looked like the shape of left hand.
And the ancient image of “右” the opposite shape.
After thousands of years, the shape of these two words have become more complicated.

If you see from its inside structure, the word “左” has a “工” inside (“工”means work). While the word “右” has a “口” inside ( “口” means month). So, in a team, “左” often means common worker who's good at basic working; while “右” often means manager who's good at speaking and arranging jobs.

About right and left which is more powerful, this is hard to say. In Chinese culture, the left represents masculine and the right represents feminine. In ancient times, most dynasties of China regarded the left as more honorable while a few dynasties regarded the right. Now in daily domestic life, Chinese regard the left as more honorable. But in international occasion, Chinese follow the global common rule and regard the right as more honorable.